Call
this operator “equivalence”, or “Einstein addition”, after relativistic
velocity addition. Denote it as (x~y). It has these laws:

(tanh x)~(tanh
y)=tanh(x+y)

x~0=x

x~(1/0)=1/x

x~1=1

x~ -1=-1

x~ -x=0

x~ -1/x=1/0

1~ -1=%

-(x~y)=(-x)~(-y)

1/(x~y)=(1/x)~y=x~(1/y)

1/(x~y)=(x {+} y) + (1/x {+} 1/y)

=(x +
1/y) {+} (1/x + y)

=(1+xy)
/ (x+y)

(1/x)~(1/y)=x~y

x~y=y~x

(x~y)~z=x~(y~z)

(x~y)~z=(x+y+z+xyz) / (xy+yz+zx+1)

=(x{+}y{+}z{+}xyz) / (xy{+}yz{+}zx{+}1)

These imply Transposition:

(A~x) = Bif and only ifx = (-A~B)

Note that multiplication and equivalence relate to
negative and reciprocal in opposite ways:

-(x*y) = (-x)*y = -1*x*y;1/(x*y) = (1/x)*(1/y)

-(x~y) =(-x)~(-y);1/(x~y) = (1/x)~y =(1/0)~x~y

Negation
distributes into multiplication and over equivalence; whereas
reciprocal distributes over multiplication and into equivalence.
To multiplication, negation is a term (-1) and reciprocal is a sign; to equivalence,
reciprocal is a term (1/0)
and negation is a sign.